Re: Notions of Type

Bob Badour wrote:
> erk wrote:
>
> > Bob Badour wrote:
> >
> >>Marshall wrote:
> >>
> >>>paul c wrote:
> >>>
> >>>>Marshall wrote:
> >>>>
> >>>>>erk wrote:
> >>>>>
> >>>>>>Sorry if this is obvious to everyone else, but does an algebra include
> >>>>>>only operations defined on values of the type in question?
> >>>>>
> >>>>>Yes.
> >>>>>
> >>>>>>I ask
> >>>>>>because in relational algebra, at least the rename operator involves a
> >>>>>>different type ("attribute name") than the "core type" (relation).
> >>>>>
> >>>>>Very true. Of the various relational operators that have been
> >>>>>identified over the years, only a few, like union, are really
> >>>>>algebraic. RESTRICT, PROJECT, etc. aren't. ...
> >>>>
> >>>>Just a quick question - is your reason for making that statement that
> >>>>restrict, project, etc. result in a relation that doesn't necessarily
> >>>>have all the attributes of the operand relations?
> >>>
> >>>No; that's actually not a problem.
> >>>
> >>>The problem is the operands. Strictly speaking, an algebraic operator
> >>>would be one which had the type
> >>>
> >>> op: 'a, 'a -> 'a
> >>>
> >>>("The thing named "op" has the type: function taking operands of
> >>>type some-a and some-a and returning a value of type some-a.")
> >>>
> >>>So the algebra of the integers has operators like:
> >>>
> >>> +: int, int -> int
> >>>
> >>>Consider PROJECT:
> >>>
> >>> PROJECT: Relation, Set-of-attributes -> Relation
> >>>
> >>>So for PROJECT of an x,y point over x, we pass it two
> >>>things:
> >>>
> >>>1) A relation defined over attributes x and y
> >>>2) ???
> >>>and it returns
> >>>3) A relation defined over attribute x (aka "profit")
> >>>
> >>>Whoops! Doesn't fit the template. The second argument isn't
> >>>a relation. So, strictly speaking, this is not an algebraic operator,
> >>>because it isn't closed over the type Relation. Exercise for the
> >>>reader: what *is* the type of the other argument? This should
> >>>make your head hurt a least a little bit.
> >>
> >>It is a relation of degree 1 and cardinality N representing a set of N
> >>attribute names.
> >
> > Must it be a relation, given that its tuples aren't used at all?
>
> The tuples are used as each tuple identifies an attribute in the result.

Ach, my apologies, I misunderstood your intent, and your comment below clarifies. I somehow read what you wrote as reversing what I understood about cardinality and degree.

> And if
> > one uses a relation, isn't the degree N and the cardinality 0? (I'm
> > basing this on TABLE_DEE (true) being degree zero, cardinality one and
> > TABLE_DUM (false) being degree zero, cardinality zero).
>
> That is an alternate way to define the operation.

Understood, thanks.

• erk